We know that won't converge, and not being able to try infinite numbers has nothing to do with it. So what you say seems to me to be, on the surface, pretty much complete nonsense. To one who is trained in mathematics, what you're saying feels very similar to people who haven't travelled much saying that the Earth is obviously flat.
With respect, this response seems needlessly harsh and supercilious. That’s the kind of stuff that discourages people from going deeper into a field. The question appears to have been asked in good faith ... why not be more charitable and constructive?
And as you probably know, it was discovered that many sequences that supposedly “won’t converge” indeed do converge under analytic continuation, there are reasonable and terribly subtle philosophical questions to ask about the validity of inductive reasoning, and there are working mathematicians who consider themselves finitists.